There are two main ways to get to the canyon. I prefer the East Mesa Trail, when a car spot is available.
East Mesa Trail:
From the East Entrance of Zion National Park, drive east 2.4 miles on highway 9. Turn left at the signed "North Fork" junction. Drive 5.2 miles to signed "Zion Ponderosa Resort". Turn left and enter the Zion Ponderosa (N37° 16' 22", W112° 52' 24"). From here, you will be driving on dirt roads.
After entering the Zion Ponderosa, stay on the main road (do not turn left to the resort) and drive 0.7 miles to signed Cable Mountain, Observation Point road fork (N37° 16' 33", W112° 53' 02"). Take the right fork and drive 0.7 miles to a tee in the road (N37° 16' 47", W112° 53' 45"). Turn right, and drive 1.1 miles until road forks around a large ponderosa pine (N37° 17' 43", W112° 53' 49"). Take the right fork and drive 0.2 miles to a spur road on the left (N37° 17' 51", W112° 54' 00"). Take the spur road to the left and drive 0.1 miles. The road ends at a fence, which is the Zion Park boundary. There are no signs at the trailhead (N37° 17' 47", W112° 54' 04").
Cross the fence at the access point 40 feet south and begin walking west on the well-traveled trail. After 20 minutes, you will notice a canyon to the north. This is the South Fork of Orderville Canyon, DO NOT drop in here. This canyon and Mystery Canyon look very similar from the top. Continue walking for an additional 20 minutes and you will again notice a canyon to the north with a spur trail leading to the edge. This is Mystery Canyon.
Directions from http://www.climb-utah.com/Zion/mystery.htm